9 – Nd113 C2 L3 09 L Simplifying The Kalman Filter Equations V3

These are the common filter equations in all their beautiful complexity. But what do they all mean.? Before we jump into all the matrix math behind these equations, I’d like to just do some basic visual cleanup. And to do that, I want to make three modifications to these equations. Let’s just focus on this first equation here so I can explain what those modifications are. The first thing I’m going to do to all these equations, is just get rid of these hats, like hats you saw at the x. While the hat does have physical significance it indicates that this variable is an estimate. It’s not for sure. It doesn’t have any mathematical significance. It’s not telling us to do anything mathematically with these equations, so just get rid of those for now. Next, we’re going to take all these x_k giving k – 1 and we’re going replace that with x’. Not only is this easier to look at. It’s also consistent with the notation you’ve been using with Sebastian. Finally we’re just going to get rid of these lone k’s, they’re cluttering things up and I’d rather just look at an equation like this. Now we have this simplified equation. Remember it’s some math here just looks simpler. Let’s take a look what happens to all of the call filter equations when we apply these same three cleaning steps. We take equations which looked like these and we convert them to equations that look like these. This representation really let’s us focus on the mathematical tools we’re going to need in order to make Wikipedia articles useful to us. In fact, we can actually use certain aspects of these equations right now to highlight what we’re going to cover in the rest lesson. First, let’s focus on the difference between lower case and upper case, there’s actually meaning there. Generally, lower case variables indicate vectors while uppercase variables indicate matrices. We’ll talk more about this distinction in a minute. We’ll also talk about addition and multiplication of vectors and matrices. We’ll discuss one particularly special matrix called the identity matrix and we’ll talk about two additional important operations called the transpose, given by this little T superscript and the Matrix inverse, given by this little -1 superscript. If this seems like an overwhelming amount of stuff to learn in a very short time, remember the goal here is not to memorize any of this. The goal isn’t even really to get a conceptual understanding of linear algebra that will come with time. The goal now is to acquire a basic functional familiarity with the tool that is matrix math, so that when you encounter it in future Google searches you’ll know where to begin.

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